1. Robot Localization Using Bayesian Methods
Stochastic Processes Mini Conference Winter 2011
EE Prof. Brian Mazzeo
Amin Nazaran
Stephen Quebe

2. Presentation Outline Robot Localization
Modeling Robot Localization as a Stochastic Process.
Bayesian Estimation and Filtering.
The Extended Kalman Filter.
Extended Kalman Filter Simulation Results.
Conclusions.

3. Robot Localization
In order for a mobile robot to complete many meaningful tasks, it must be able to identify and control its position in an environment.
“Using sensory information to locate the robot in its environment is the most fundamental problem in robotics [1].”

4. The Localization Problem
Given a map of an environment and a sequence of sensor measurements and control inputs, estimate the robot’s pose.

5. The Localization Problem
Inputs
Outputs
Robot initial pose.
Control inputs.
Observations.
Map feature or landmarks.
Estimated robot pose.

6. Robot Motion and Observation Models

7. Modeling Robot Localization as a Stochastic Process
One approach to solving this problem is by modeling the robot’s control inputs, observations using a Markov Chain.

8. The Markov Assumption
The Markov assumption states that if we know the current state of the robot, past and future states are conditionally independent of one another.
In other words. If we know where the robot is now, then knowing where the robot was 5 minutes ago doesn’t give us any more information than we already have, regarding it’s current state.
The arrows on Dynamic Bayes Network show this conditional independence.

9. Stochastic Motion Model
The robot motion model describes the robot’s pose as a function of it’s previous pose and control inputs.
The observation model describes the robot’s sensor measurements as a function of the robot’s position and the landmark position.

10. Stochastic Motion Model Bayes Network

11. Bayesian Estimation and Filtering
It is a recursive algorithm. At time t, given the belief at time t-1 belt-1(xr), the last motion control ut-1 and the last measurement zt, determine the new belief belt(xr) as follows:
Motion model
Measurement model

12. Bayesian Estimation: Prediction
Motion model
Robot pose space
Based on the total probability theorem:
(discrete case)
where Bi, i=1,2,... is a partition of W. In the continuous case:

13. The Extended Kalman Filter (EKF)
The Extended Kalman Filter is one way to apply Bayesian estimation techniques to robot localization and mapping.
The Kalman filter is the optimal Least Mean Squares estimator of a linear Gaussian system.
The Extended Kalman filter is a way of using the Kalman filter with non-linear models by approximating the model.

14. EKF Assumptions and Violations
Gaussian noise and uncertainty.
Linear approximations are good.
Markov assumption or complete state assumption holds.
Violations:
Data association create Non-Gaussian uncertainties.
With large time steps or angles the linear approximation is poor.
If the estimate becomes unstable or overconfident the Markov assumption is violated by a poor estimate.
If the robot is “bumped” or moved by something not in the model, the Markov is also violated.

15. EKF Assumptions and Violations

16. EKF Algorithm EKF_localization ( mt-1, St-1, ut, zt, m): Prediction:
Jacobian of g w.r.t location
Jacobian of g w.r.t control

17. EKF Algorithm Continued17
Motion noise
Predicted mean
Predicted covariance 17

18. EKF Measurement Update
Normalizing factor
Measurement model
Based on the Bayes Rule:
i.e. also:
Taking:
We have:

19. EKF_localization ( mt-1, St-1, ut, zt, m): Correction:
Predicted measurement mean
Jacobian of h w.r.t location
Pred. measurement covariance
Kalman gain
Updated mean
Updated covariance

20. EKF Simulation Results
Normal operation.
Overconfident prediction.
Overconfident measurement.
Large time steps where linearization fails.
External bump where Markov assumption fails.

21. Simulation Results
Show simulation results in real time by opening matlab.

22. Conclusions
The critical assumption in the stochastic model is the Markov assumption. This assumption is restrictive but probably cannot be avoided in any real world scenario.
The Extended Kalman Filter implementation is fast and remains consistent under normal conditions.
In the real world the model can be adjusted to reduce and recover from failure.
The robot must be able to recognize and recover from inevitable failure (the lost robot problem).

23. Thank You For Your Attention Questions?

24. References
[1]: I.J. Cox. Blanche—an experiment in guidance and navigation of an autonomous robot vehicle. IEEE Transactions on Robotics and Automation, vol.7,NO.2 ,pp.193–204, [2] S. Thrun,W. Burgard, and D.Fox, “Probabilistic Robotics”, MIT press: Cambridge, 1967.